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5.2 arithmetic sequences

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5.2 arithmetic sequences

1. 1. Arithmetic Sequences
2. 2. Arithmetic SequencesA sequence a1, a2 , a3 , … is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.
3. 3. Arithmetic SequencesA sequence a1, a2 , a3 , … is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, …is an arithmetic sequence because an = 2n – 1.
4. 4. Arithmetic SequencesA sequence a1, a2 , a3 , … is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, …is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 – ak = d.
5. 5. Arithmetic SequencesA sequence a1, a2 , a3 , … is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, …is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 – ak = d.In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.
6. 6. Arithmetic SequencesA sequence a1, a2 , a3 , … is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, …is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 – ak = d.In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.The following theorem gives the converse of the above factand the main formula for arithmetic sequences.
7. 7. Arithmetic SequencesA sequence a1, a2 , a3 , … is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, …is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 – ak = d.In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.The following theorem gives the converse of the above factand the main formula for arithmetic sequences.Theorem: If a1, a2 , a3 , …an is a sequence such thatan+1 – an = d for all n, then a1, a2, a3,… is an arithmeticsequence
8. 8. Arithmetic SequencesA sequence a1, a2 , a3 , … is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, …is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 – ak = d.In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.The following theorem gives the converse of the above factand the main formula for arithmetic sequences.Theorem: If a1, a2 , a3 , …an is a sequence such thatan+1 – an = d for all n, then a1, a2, a3,… is an arithmeticsequence and the formula for the sequence isan = d(n – 1) + a1.
9. 9. Arithmetic SequencesA sequence a1, a2 , a3 , … is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, …is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 – ak = d.In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.The following theorem gives the converse of the above factand the main formula for arithmetic sequences.Theorem: If a1, a2 , a3 , …an is a sequence such thatan+1 – an = d for all n, then a1, a2, a3,… is an arithmeticsequence and the formula for the sequence isan = d(n – 1) + a1. This is the general formula of arithemeticsequences.
10. 10. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.
11. 11. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence.
12. 12. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
13. 13. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.b. Find the (specific) formula represents this sequence.
14. 14. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1
15. 15. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1 we get an = 3(n – 1) + 2
16. 16. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1 we get an = 3(n – 1) + 2 an = 3n – 3 + 2
17. 17. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1 we get an = 3(n – 1) + 2 an = 3n – 3 + 2 an = 3n – 1 the specific formula.
18. 18. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1 we get an = 3(n – 1) + 2 an = 3n – 3 + 2 an = 3n – 1 the specific formula.c. Find a1000.
19. 19. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1 we get an = 3(n – 1) + 2 an = 3n – 3 + 2 an = 3n – 1 the specific formula.c. Find a1000. Set n = 1000 in the specific formula,
20. 20. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, …a. Verify it is an arithmetic sequence. Its arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1 we get an = 3(n – 1) + 2 an = 3n – 3 + 2 an = 3n – 1 the specific formula.c. Find a1000. Set n = 1000 in the specific formula, we get a1000 = 3(1000) – 1 = 2999.
21. 21. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.
22. 22. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.
23. 23. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1,
24. 24. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.
25. 25. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula,
26. 26. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5
27. 27. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5
28. 28. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25
29. 29. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1
30. 30. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1an = -4(n – 1) + 25
31. 31. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1an = -4(n – 1) + 25an = -4n + 4 + 25
32. 32. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1an = -4(n – 1) + 25an = -4n + 4 + 25an = -4n + 29
33. 33. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1an = -4(n – 1) + 25an = -4n + 4 + 25an =find a1000, set n = 1000 in the specific formulaTo -4n + 29
34. 34. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = –4 in the general formula an = d(n – 1) + a1, we getan = –4(n – 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1an = -4(n – 1) + 25an = -4n + 4 + 25an =find a1000, set n = 1000 in the specific formulaTo -4n + 29a1000 = –4(1000) + 29 = –3971
35. 35. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.
36. 36. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1,
37. 37. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we geta3 = d(3 – 1) + a1 = -3
38. 38. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we geta3 = d(3 – 1) + a1 = -3 2d + a1 = -3
39. 39. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 2d + a1 = -3
40. 40. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3
41. 41. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3
42. 42. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42
43. 43. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d=7
44. 44. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d=7Put d = 7 into 2d + a1 = -3,
45. 45. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d=7Put d = 7 into 2d + a1 = -3,2(7) + a1 = -3
46. 46. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d=7Put d = 7 into 2d + a1 = -3,2(7) + a1 = -314 + a1 = -3
47. 47. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d=7Put d = 7 into 2d + a1 = -3,2(7) + a1 = -314 + a1 = -3a1 = -17
48. 48. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d=7Put d = 7 into 2d + a1 = -3,2(7) + a1 = -314 + a1 = -3a1 = -17
49. 49. Arithmetic SequencesExample D. Given that a1, a2 , a3 , …is an arithmetic sequencewith a3 = -3 and a9 = 39, find d, a1 and the specific formula.Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a9 = d(9 – 1) + a1 = 39a3 = d(3 – 1) + a1 = -3 8d + a1 = 39 2d + a1 = -3Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d=7Put d = 7 into 2d + a1 = -3,2(7) + a1 = -314 + a1 = -3a1 = -17
50. 50. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n
51. 51. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n Head Tail
52. 52. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2
53. 53. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2Example E.a. Given the arithmetic sequence a1= 4, 7, 10, … , andan = 67. What is n?
54. 54. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2Example E.a. Given the arithmetic sequence a1= 4, 7, 10, … , andan = 67. What is n?We need the specific formula.
55. 55. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2Example E.a. Given the arithmetic sequence a1= 4, 7, 10, … , andan = 67. What is n?We need the specific formula. Find d = 7 – 4 = 3.
56. 56. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2Example E.a. Given the arithmetic sequence a1= 4, 7, 10, … , andan = 67. What is n?We need the specific formula. Find d = 7 – 4 = 3.Therefore the specific formula isan = 3(n – 1) + 4
57. 57. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2Example E.a. Given the arithmetic sequence a1= 4, 7, 10, … , andan = 67. What is n?We need the specific formula. Find d = 7 – 4 = 3.Therefore the specific formula isan = 3(n – 1) + 4an = 3n + 1.
58. 58. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2Example E.a. Given the arithmetic sequence a1= 4, 7, 10, … , andan = 67. What is n?We need the specific formula. Find d = 7 – 4 = 3.Therefore the specific formula isan = 3(n – 1) + 4an = 3n + 1.If an = 67 = 3n + 1,
59. 59. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2Example E.a. Given the arithmetic sequence a1= 4, 7, 10, … , andan = 67. What is n?We need the specific formula. Find d = 7 – 4 = 3.Therefore the specific formula isan = 3(n – 1) + 4an = 3n + 1.If an = 67 = 3n + 1, then 66 = 3n
60. 60. Sum of Arithmetic SequencesGiven that a1, a2 , a3 , …an an arithmetic sequence, then ( Head 2 Tail ) +a1+ a2 + a3 + … + an = n a1 + an = n( ) Head Tail 2Example E.a. Given the arithmetic sequence a1= 4, 7, 10, … , andan = 67. What is n?We need the specific formula. Find d = 7 – 4 = 3.Therefore the specific formula isan = 3(n – 1) + 4an = 3n + 1.If an = 67 = 3n + 1, then 66 = 3n or 22 = n
61. 61. Sum of Arithmetic Sequencesb. Find the sum 4 + 7 + 10 +…+ 67
62. 62. Sum of Arithmetic Sequencesb. Find the sum 4 + 7 + 10 +…+ 67a1 = 4, and a22 = 67 with n = 22,
63. 63. Sum of Arithmetic Sequencesb. Find the sum 4 + 7 + 10 +…+ 67a1 = 4, and a22 = 67 with n = 22, so the sum 4 + 674 + 7 + 10 +…+ 67 = 22 ( 2 )
64. 64. Sum of Arithmetic Sequencesb. Find the sum 4 + 7 + 10 +…+ 67a1 = 4, and a22 = 67 with n = 22, so the sum 11 4 + 674 + 7 + 10 +…+ 67 = 22 ( 2 )
65. 65. Sum of Arithmetic Sequencesb. Find the sum 4 + 7 + 10 +…+ 67a1 = 4, and a22 = 67 with n = 22, so the sum 11 4 + 674 + 7 + 10 +…+ 67 = 22 ( 2 ) = 11(71)
66. 66. Sum of Arithmetic Sequencesb. Find the sum 4 + 7 + 10 +…+ 67a1 = 4, and a22 = 67 with n = 22, so the sum 11 4 + 674 + 7 + 10 +…+ 67 = 22 ( 2 ) = 11(71) = 781
67. 67. Arithmetic Sequences
68. 68. Sum of Arithmetic SequencesFind the specific formula then the arithmetic sum.a. – 4 – 1 + 2 +…+ 302b. – 4 – 9 – 14 … – 1999c. 27 + 24 + 21 … – 1992d. 3 + 9 + 15 … + 111,111,111